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Chapter 11: Q11-16E (page 653)

Construct a scatterplot for the data in the following table.

X

.5

1

1.5

y

2

1

3

a. Plot the following two lines on your scatterplot: y = 3 - x and y = 1 + x

b. Which of these lines would you choose to characterize the relationship between x and y? Explain.

c. Show that the sum of the prediction errors for both of these lines equals 0.

d. Which of these lines has the smaller SSE?

e. Determine the data's least-squares line and compare it to the two lines described in part a.

Short Answer

Expert verified

Answer

  1. Fig. 1 Scatterplot
  2. Fig. 2 Table
  3. Fig. 3 Table
  4. Fig. 4 Table, Fig. 2 Table
  5. Fig. 6 Table

Step by step solution

01

Introduction

The total variance of the response values from the match to the response values is calculated by this statistic. The sum square of regression is another name for it, and it's denoted as SSE.

02

(a) Draw a scatterplot

03

(b) The relationship between x and y

The lines I used to characterize the relation between x and y are:

Y = 1 + x

The data is compared to a close-ratio of x and y.

04

(c) Show that the sum of the prediction errors for both of these lines equals 0

The sum of the prediction errors for both of these lines is:

Therefore, the sum of the prediction errors for both lines equals 0.

05

(d) Find which line has a smaller SSE

y = 3−x

y = 1 + x

We may derive SSE from the two tables above:

∑(y−y¯)2=0.25

Therefore, both the lines have the same SSE.

06

(e) Determine the data's least-squares line and compare it to the two lines described in part a

The least-square line is Y = 1 + x.

From part the line equation, i.e., Y = 1 + xequivalent to the given data.

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Most popular questions from this chapter

Generation Y’s entitlement mentality. The current workforce is dominated by “Generation Y”—people born between 1982 and 1999. These workers have a reputation as having an entitlement mentality (e.g., they believe they have a right to a high-paying job, without the work ethic). The reasons behind this phenomenon were investigated in Proceedings of the Academy of Educational Leadership (Vol. 16, 2011). A sample of 272 undergraduate business students was administered a questionnaire designed to capture the behaviors that lead to an entitlement mentality. The responses were used to measure the following two quantitative variables for each student: entitlement score (y)—where higher scores indicate a greater level of entitlement, and “helicopter parents” score (x)—where higher scores indicate that the student’s parents had a higher level of involvement in his or her everyday experiences and problems.

a. Give the equation of a simple linear regression model relating y to x.

b. The researchers theorize that helicopter parents lead to an entitlement mentality. Based on this theory, would you expect β0 to be positive or negative (or are you unsure)? Would you expect β1 to be positive or negative (or are you unsure)? Explain.

c. The p-value for testing H0: β0 = 0 versus Ha: β1> 0 was reported as .002. Use this result to test the researchers’ entitlement theory at α = .01.

Voltage sags and swells. The power quality of a transformer is measured by the quality of the voltage. Two causes of poor power quality are "sags" and "swells." A sag is an unusual dip and a swell is an unusual increase in the voltage level of a transformer. The power quality of transformers built in Turkey was investigated in Electrical Engineering (Vol. 95, 2013). For a sample of 103 transformers built for heavy industry, the mean number of sags per week was 353 and the mean number of swells per week was 184. Assume the standard deviation of the sag distribution is 30 sags per week and the standard deviation of the swell distribution is 25 swells per week.

a. For a sag distribution with any shape, what proportion of transformers will have between 263 and 443 sags per week? Which rule did you apply and why?

b. For a sag distribution that is mound-shaped and symmetric, what proportion of transformers will have between 263 and 443 sags per week? Which rule did you apply and why?

c. For a swell distribution with any shape, what proportion of transformers will have between 109 and 259 swells per week? Which rule did you apply and why?

d. For a swell distribution that is mound-shaped and symmetric, what proportion of transformers will have between 109 and 259 swells per week? Which rule did you apply and why?

Minitab was used to generate the following histogram:

a. Is this a frequency histogram or a relative frequency histogram? Explain.

b. How many measurement classes were used in the construction of this histogram?

c. How many measurements are in the data set described by this histogram?

State Math SAT scores. Refer to the simple linear regression relating y = 2014 Math SAT scores to x = 2010 Math SAT scores, Exercise 11.19 (p. 654). A portion of the SPSS printout of the analysis is shown below.

a. Locate the values of SSE, s2, and s on the SPSS printout.

b. Give a practical interpretation of the value of s.

The following table is similar to Table 11.2. It is used for making the preliminary computations for finding the least-squares line for the given pairs of x- and y-values.

a. Complete the table.

b. FindSSxy.

c. Find SSxx.

d. Findβ1^.

e. FindX¯andY¯.

f. Findβ0^.

g. Find the least-squares line.
See all solutions

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